SEMINARIO: Recent Progress on Derivative-Free Trust-Region Methods

Data dell'evento:

Martedì, 28 Giugno, 2016 - 14:30

Luogo:

Aula A4 - DIAG Via Ariosto

Speaker:

Prof. Luis Nunes Vicente, University of Coimbra

Abstract

Trust-region methods are a broad class of methods for continuousoptimization that finds application in a variety of problemsand contexts. The basic principle consists of iteratively optimizinga model of the objective function in a restricted region. In particular,they have been studied and applied for problems without using derivatives,where models are built solely by sampled function values.

Trust-region derivative-free methods are guaranteed toconverge for deterministic smooth functions, in the senseof generating a sequence or run of iterates converging to criticality(of first and second order type). Such a convergence is calledglobal as there is no assumption on the starting point.It has also been proved that the order of complexity, or theglobal rate in which criticality decays, matches thederivative-based case.

In the deterministic non-smooth case, and given some knowledgeof the non-smooth structure, it is possible to design globallyconvergent approaches with appropriated complexity, either bysmoothing the original function in some parametrized way or bymoving a compositive non-smooth structure directly to thetrust-region subproblem.

Trust-region methods can be based as well on probabilistic modelsof the objective function, thus considering the derivative-freecase where sample points are randomly generated. Such methodsexhibit similar properties of convergence and complexity as thoseusing deterministic models, now not for all runs but for aa set of those that occurs with probability one.

Finally, we are observing now the first attempts for thestochastic case, where the objective function can only beobserved, and possibly approximated by sample averaging.

This talk will attempt to overview all such developments andpoint out what still remains unknown.